Trigonometric substitutions

I received the following email:
I can't see the logic in assuming (for example) that a function containing sqrt (a^2 - x^2) in the integrand would lead you to substitute x with a trig reference. Why a trig reference? What connection does the integrand, trig-less function have to trigonometry? To me, it seems about as rational as replacing the 'a' with pi, or e, or a sausage!
How can I make this tangible?
Are there any online illustrations of this?

Let him substitute x=a cos(t) and see for himself. Remember sin^2(t)+cos^2(t)=1. The squares in this equation are the link to square roots.
It's pretty self-explanatory once you've actually tried it out.